# Dynamic Uncertain Causality Graph Applied to the Intelligent Evaluation of a Shale-Gas Sweet Spot

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Causality Representation

_{6}in Figure 1, with its state expression specified in Figure 2. A D-type variable (pentagon node) represents the default or unspecified cause of an X-type variable. Variable V

_{i,j}represents a parent event of X

_{n}, V∈{B, X, G, D}, n indexes the child variable X, k denotes the variable’s state, i indexes the parent variable V, and j denotes the variable’s state. State j = 0 indicates an uninterested state, and j ≠ 0 indicates an interested state.

_{nk;ij}≡ (r

_{n;i}/r

_{n})A

_{nk;ij}describes the causality between a child event X

_{nk}and a parent event V

_{ij}, and A

_{nk;ij}denotes the independent causality function that V

_{ij}does cause X

_{nk}. The weighting factor r

_{n;i}represents the intensity of causality between X

_{n}and V

_{i}, where r

_{n;I}> 0 and ${r}_{n}\equiv {\displaystyle \sum}_{i}{r}_{n}$. As a weighting coefficient of A

_{n;i}, r

_{n;i}/r

_{n}balances these independent causality functions for all parent events on the same child event.

_{n;i}between a child event X

_{n}and a parent event V

_{i}is met, the dashed arc is converted into a solid arc; otherwise, it is deleted. In the DUCG, the variable identifiers in lowercase letters denote the probability parameters of variables in corresponding uppercase letters, for example, b

_{ij}= Pr{B

_{ij}}, a

_{nk}

_{;ij}= Pr{A

_{nk}

_{;ij}}.

#### 2.2. Probabilistic Reasoning

_{nk}can be expanded into the sum-of-product expression of a series of independent events according to its causality, as shown in Equation (1); the event expansion of the DUCG is shown in Figure 3 [22].

_{kj}is the hypothesis event to be solved that is composed of {B, X, A}-type events. The reasoning process for the DUCG is as follows:

- 1.
- Simplification

- 2.
- Decomposition

_{i}, the DUCG simplified in step 1 can be divided into a set of sub-DUCGs, which can be further simplified according to the rules. B

_{i}is added to the hypothetical space S

_{H}.

- 3.
- Event outspread

_{j}(E = ∏

_{j}E

_{j}); by combining them with logical operations, the final expression of the E expansion is obtained by combining {B, D, A, r}-type events and parameters. Similarly, H

_{kj}E can be expanded to an expression containing only {B, D, A, r} types of events and parameters.

- 4.
- Probabilistic calculation

_{kj}denotes possible hypothetical events, and Pr{E} and Pr{H

_{kj}E} in each sub-DUCG can be calculated. For the whole simplified DUCG, the probability calculation includes state- and ranking-probability calculations. The formula of the state-probability calculation is as follows:

^{s}

_{kj}is the posterior probability of H

_{kj}; the ranking probability of H

_{kj}is as follows:

#### 2.3. Extension of DUCG

#### 2.3.1. Multiple Conditional Events

_{n;i}, but each evaluation indicator in the evaluation of oil and gas reservoir sweet spots is in more than one state and the relationship among the evaluation indicators is more complicated. It is difficult to clearly represent the relationship between the parent variable V

_{i}and the child variable X

_{n}with a conditional event Z

_{n;i}. This paper introduced multiple conditional events Z

_{nk;ij}to expand Z

_{n}

_{;i}. Z

_{nk;ij}denotes the condition when state j of the parent variable V

_{i}and state k of the child variable X

_{n}satisfy causality. As illustrated in Figure 4, F

_{3;1}is a conditional causality. Without losing generality, suppose there are two states for each variable, states 0 and 1, and the condition is Z

_{3,1;1,1}= X

_{2,0}; i.e., only when X

_{2}is in state 0 does F

_{3,1;1,1}exist; otherwise, F

_{3,1;1,1}is eliminated.

#### 2.3.2. Weighted Graph

_{k}denotes the weighted graph generated according to each state k of the unknown variable, and ω

_{i}denotes the probability of the weighted graph indexed by i. Because the weighted graphs may have different structures, it was necessary to determine each weighted graph before reasoning and then reason in each weighted graph in a conventional way: (1) simplification, (2) decomposition, (3) event outspread, and (4) probabilistic calculation.

_{i}denotes evidence E in DUCG

_{i}.

## 3. Evaluation Model Construction

- 1.
- Variable definition

- 2.
- Knowledge representation

- 3.
- Probability reasoning

- 4.
- Comprehensive comparison

#### 3.1. Screening Critical Factors for Shale-Gas Exploration and Development

^{2}. The favorable area and the organic matter thickness are combined to ensure the entire shale-gas sweet-spot content. The gas content of shale-gas reservoirs is the most direct reflection of the economic value of the target area, and its value in the target area should be greater than 4 m

^{3}/t. The tectonic background plays a key role in the development of shale-gas reservoirs and creates corresponding requirements for the development technology of shale gas.

#### 3.2. Establishment of the Evaluation Model for Shale-Gas Sweet Spots

#### 3.2.1. Define Variables

#### 3.2.2. Determine Causality

_{1,1}), the area must be simultaneously the best geological (X

_{4,2}), engineering (X

_{5,2}), and economic (X

_{6,2}) sweet spots and all must satisfy certain conditions. In the DUCG, conditional events that are represented by directed arcs with red dashed lines were used to represent this type of constraint relationship, i.e., whether the causality connection between variables was established depended on other conditional judgments.

_{1,1}). The original conditional variable in the DUCG can only use Z

_{4;1}to represent the conditional event between the evaluation target area (B

_{1}) and the geological sweet spot (X

_{4}), but Z

_{4;1}does not represent the two conditional events between the shale-gas evaluation target area (B

_{1,1}) and the geological desert area with the general level X

_{4,1}and the better level X

_{4,2}. Therefore, multiple conditional events Z

_{nk;ij}are introduced between the parent variable V

_{i}and the child variable X

_{n}and the conditional events of B

_{1,1}are extended as follows: the conditional event between B

_{1,1}and X

_{4,2}is Z

_{4,2;1,1}= X

_{5,2}X

_{6,2}+ X

_{5,1}X

_{6,2}+ X

_{5,2}X

_{6,1}. As shown in Figure 7, after the conditions were satisfied, the dashed red line in Figure 6 became a solid red line, with blue representing the variable in state 1 and yellow representing the variable in state 2.

#### 3.2.3. Determine Causality Function Parameters

_{kj}|E} = Pr{H

_{kj}E}/Pr{E}, where the posterior probability of H

_{kj}depends on the relative values of Pr{H

_{kj}E} and Pr{E}. As a result, the accuracy of {a-, r-}-type parameters has only relative meaning. Therefore, it is realistic for domain experts to specify {a-, r-}-type parameters of the DUCG directly based on their knowledge in cases without statistic data.

_{i}and the child variable X

_{n}, so the intensity of causality r

_{n;i}is 1. The initial probability of each B-type variable is the same. The given causality function parameters are shown in Figure 8, in which “−” means that this causality is not of concern or there is no contribution from this parent event. The other values in the matrix represent the causality between parent and child variables. For example, a

_{4,2;1,1}= 0.7, which means that, when the area is a shale-gas target area (B

_{1,1}), the probability of it being a good geological desert area (x

_{4,2}) is 70%.

## 4. Results and Discussion

#### 4.1. Results from Complete Data

_{1}, E

_{2}, E

_{3}, and E

_{4,}respectively, collectively referred to as evidence E. With evidence E, the state of each variable in Figure 6 and the conditional event could be simultaneously determined. After simplification, the final evaluation model was obtained. A shale-gas sweet spot was then evaluated through the DUCG reasoning algorithm. The evaluation result was compared with the evaluation results of other models to verify the effectiveness of the model.

_{1}is simplified, i.e., the evaluation results of Changning and Jiaoshiba were shale-gas exploration and development target areas (level I areas).

_{1}and B

_{3}are simplified, respectively. The possible hypotheses were calculated as given below.

_{2,1}= B

_{2,1}and H

_{1,1}= B

_{1,1}; therefore,

#### 4.2. Results from Incomplete Data

_{12}) is unknown. In this case, the sweet spot was evaluated when the evidence was incomplete.

_{5}= E

_{8,4}E

_{9,2}E

_{10,1}E

_{11,1}E

_{13,3}E

_{14,3}E

_{15,2}E

_{16,2}E

_{17,3}E

_{18,2}E

_{19,2}and the conditional event in Table 4 could be converted from Figure 6 to Figure 11. The state of variable X

_{9}in Figure 11 was unknown, so the conditional event between X

_{5}and its child variables could not be determined. In the DUCG, if the state of the variable in the conditional event is unknown, the graph structure cannot be determined. At this time, the prior probability of the effective state of the variable must be given or calculated in some way. Note that if state k of X

_{12}is given, the state of X

_{5}and the conditional events with child variables X

_{11}, X

_{12}, X

_{13}, and X

_{14}can be determined. Similarly, whether the conditional event between B

_{1}, B

_{2}, B

_{3}and X

_{4}, X

_{5}, X

_{6}is true and the entire evaluation model can also be determined. According to E

_{5}, Pr{X

_{12,k}} was calculated, which is more objective than directly providing Pr{X

_{12,k}}. When the conditional event Z is not observed, the prior distribution of Z can be expanded in the form of a full combination [22]. However, when the graph results are complex and there are many conditional events, this method expression is not clear enough and the calculation is large.

_{12}has three states in Table 3; for each state, the evaluation model was different (or the same, the specific depending on the conditional event). The weighted graphs were generated according to each state of X

_{12}. For each weighted graph, the possible evaluation results were obtained, and then the evaluation results of all weighted graphs were added to obtain the total evaluation result. If there were multiple variables in the condition event whose states were unknown, their possible states were combined and each state in the state combination then corresponded to a weighted graph. Lastly, the possible evaluation results were produced according to each weighted graph.

_{5}as an example to show the calculation method in the case of subgraphs.

_{5}, the state of X

_{12}is not given and there are three possible states for X

_{12}to be considered. This needs to be divided into three subgraphs. When X

_{12}is in state 1, DUCG

_{1}denotes subgraph 1 and ω

_{1}= Pr{X

_{12,1}} denotes the probability of DUCG

_{1}. When X

_{12}is in state 2, DUCG

_{2}denotes subgraph 2 and ω

_{2}= Pr{X

_{12,2}} denotes the probability of DUCG

_{2}. When X

_{12}is in state 3, DUCG

_{3}denotes subgraph 3 and ω

_{3}= Pr{X

_{12,3}} denotes the probability of DUCG

_{3}. The divided subgraphs are shown in Figure 12.

_{1}is isolated under evidence E

_{5}because certain conditional events are not true. In Figure 12b,c, B

_{3}is isolated and these variables are deleted under the simplified DUGG rule. The reasoning process of the DUCG adds the operation of dividing the weighted graph before it is simplified in the case of incomplete data. According to Equation (4), in Figure 12, E

_{5}is expanded in DUCG

_{1}, DUCG

_{2}, and DUCG

_{3}:

_{1}, we have:

_{2}, we have:

_{3}, we have:

_{kj}is ranked by probability:

## 5. Conclusions

_{nk;ij}were proposed to satisfy the complex causality between evaluation indicators and (2) a weighted graph was created for the uncertain evidence of conditional events and comprehensive multiple-expert knowledge. Finally, the evaluation model was verified using data examples of typical shale-gas sweet spots. The results demonstrated that compared with other methods, the evaluation model based on the DUCG does not depend on data, its reasoning results are accurate, and the reasoning process can be graphically presented, making its conclusion more objective, credible, and explanatory.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Child variable combination of target area evaluation: (

**a**) Z

_{4,2;1,1}= X

_{5,2}X

_{6,2}; (

**b**) Z

_{4,2;1,1}= X

_{5,1}X

_{6,2}; (

**c**) Z

_{4,2;1,1}= X

_{5,2}X

_{6,1}.

**Figure 8.**Causality function parameters of the variables in Figure 6.

**Figure 9.**Evaluation models of shale-gas sweet spots with E in (

**a**) Changning with E

_{1}, (

**b**) Weiyuan with E

_{2}, (

**c**) Fushun-Yongchuan with E

_{3}, and (

**d**) Jiaoshiba with E

_{4}.

**Figure 10.**Simplified evaluation model of shale-gas sweet spots in (

**a**) Changning with E

_{1}, (

**b**) Weiyuan with E

_{2}, (

**c**) Fushun-Yongchuan with E

_{3}, and (

**d**) Jiaoshiba with E

_{4}.

**Figure 12.**Subgraphs of X

_{12}in different states with E

_{5}, listed as (

**a**) DUCG

_{1}, (

**b**) DUCG

_{2}, and (

**c**) DUCG

_{3}.

Relevant Industry Standard | Standard Number | State |
---|---|---|

Method of geological evaluation of natural-gas reserves | SY/T 5601-2009 | Current |

Technical requirements for economic evaluation of gas field development and adjustment programs | SY/T 6177-2009 | Current |

Evaluation method of oil and gas reservoirs | SY/T 6285-2011 | Current |

Sweet Spot | Evaluation Indicator | Target Area | Favorable Area | Prospecting Area |
---|---|---|---|---|

Geological sweet spot | Thickness of organic-rich shale/m | >50 | 50–30 | <30 |

Total organic carbon (TOC)/% | >4 | 4–2 | <2 | |

Maturity of organic matter (Ro)/% | 1.1–2.5 | 2.5–4 | >4 | |

Favorable area/km^{2} | >2500 | 2500–1000 | <1000 | |

Gas content/(m^{3}·t^{−1}) | >4 | 4–2 | <2 | |

Tectonic setting | Anticline normal structure | Slope | Syncline negative structure | |

Engineering sweet spot | Burial depth/km | 1.5–3.5 | 0.5–1.5 | <0.5 or 3.5–4.5 |

Pressure factor | >1.3 | 1.0–1.3 | <1.0 | |

Degree of natural cracks | Full development | Interlayer seam development | No development | |

Surface condition | Plains and hills | Mountains or dams | Lakes or valleys | |

Economic sweet spot | Market demand | Demand exceeds supply | Balance between supply and demand | Supply exceeds demand |

Infrastructure | In a pipe network | Close to a pipe network | Away from a pipe network |

Name | Description | State and State Description |
---|---|---|

B_{1} | Shale-gas target area | 0: false; 1: true |

B_{2} | Shale-gas favorable area | 0: false; 1: true |

B_{3} | Shale-gas prospective area | 0: false; 1: true |

X_{4} | Geological sweet spot | 0: false; 1: common; 2: good |

X_{5} | Engineering sweet spot | 0: false; 1: common; 2: good |

X_{6} | Economic sweet spot | 0: false; 1: common; 2: good |

X_{7} | Organic matter | 0: unknown; 1: common; 2: good |

X_{8} | Favorable area (km^{2}) | 0: unknown; 1: <1000; 2: >1000 and <2500; 3: >2500 |

X_{9} | Gas content (m^{3}·t^{−1}) | 0: unknown; 1: <2; 2: 2–4; 3: >4 |

X_{10} | Tectonic setting | 0: unknown; 1: syncline negative structure; 2: slope; 3: anticline normal structure |

X_{11} | Burial depth (km) | 0: unknown; 1: <0.5 or 3.5–4.5; 2: 0.5–1.5; 3: 1.5–3.5 |

X_{12} | Pressure factor | 0: unknown; 1: <1.0; 2: 1.0–1.3; 3: >1.3 |

X_{13} | Degree of natural cracks | 0: unknown; 1: undeveloped; 2: interlayer seam development; 3: fully developed |

X_{14} | Surface condition | 0: unknown; 1: lakes or valleys; 2: mountains or dams; 3: plains and hills |

X_{15} | Market demand | 0: unknown; 1: supply exceeds demand; 2: balanced; 3: demand exceeds supply |

X_{16} | Infrastructure | 0: unknown; 1: away from a pipe network; 2: close to a pipe network; 3: in a pipe network |

X_{17} | Thickness of organic-rich shale (m) | 0: unknown; 1: <30; 2: 30–50; 3: >50 |

X_{18} | Total organic carbon (%) | 0: unknown; 1: <2; 2: 2–4; 3: >4 |

X_{19} | Organic maturity (%) | 0: unknown; 1: >4; 2: 2–4; 3: 1.1–2.5 |

No. | Parent Variable | List or Description of Condition Events |
---|---|---|

1 | B_{1,1} | Z_{4,2;1,1} = X_{5,2}X_{6,2} + X_{5,1}X_{6,2} + X_{5,2}X_{6,1}Z _{4,1;1,1} = X_{5,2}X_{6,2}Z _{5,2;1,1} = X_{4,2}X_{6,2} + X_{4,2}X_{6,1} + X_{4,1}X_{6,2}Z _{5,1;1,1} = X_{4,2}X_{6,2}Z _{6,2;1,1} = X_{4,2}X_{5,2} + X_{4,2}X_{5,1} + X_{4,1}X_{5,2}Z _{6,1;1,1} = X_{4,2}X_{5,2} |

2 | B_{2,1} | Z_{4,2;2,1} = X_{5,1}X_{6,2} + X_{5,2}X_{6,1} + X_{5,1}X_{6,1}Z _{4,1;2,1} = X_{5,2}X_{6,2} + X_{5,2}X_{6,1} + X_{5,1}X_{6,2}Z _{5,2;2,1} = X_{4,2}X_{6,1} + X_{4,1}X_{6,2} + X_{4,1}X_{6,1}Z _{5,1;2,1} = X_{4,2}X_{6,2} + X_{4,2}X_{6,1} + X_{4,1}X_{6,2}Z _{6,2;2,1} = X_{4,2}X_{5,1} + X_{4,1}X_{5,2} + X_{4,1}X_{5,1}Z _{6,1;2,1} = X_{4,2}X_{5,2} + X_{4,2}X_{5,1} + X_{4,1}X_{5,2} |

3 | B_{3,1} | Z_{4,1;3,1} = X_{5,1}X_{6,1} + X_{5,2}X_{6,1} + X_{5,1}X_{6,2} Z_{4,2;3,1} = X_{5,1}X_{6,1}Z _{5,1;3,1} = X_{4,1}X_{6,1} + X_{4,1}X_{6,2} + X_{4,2}X_{6,1} Z_{5,2;3,1} = X_{4,1}X_{6,1}Z _{6,1;3,1} = X_{4,1}X_{5,1} + X_{4,1}X_{5,2} + X_{4,2}X_{5,1} Z_{6,2;3,1} = X_{4,1}X_{5,1} |

4 | X_{4,2} | All child variables of X_{4} are in state 2 or 3. |

5 | X_{4,1} | ${Z}_{4,1}=\overline{{Z}_{4,2}}$ |

6 | X_{5,2} | All child variables of X_{5} are in state 2 or 3, or X_{11} is state 1, and two of the other three variables are in state 3. |

7 | X_{5,1} | ${Z}_{5,1}=\overline{{Z}_{5,2}}$ |

8 | X_{6,2} | All child variables of X_{6} are in state 2 or 3. |

9 | X_{6,1} | ${Z}_{6,1}=\overline{{Z}_{6,2}}$ |

10 | X_{7,2} | All child variables of X_{7} are in state 2 or 3. |

11 | X_{7,1} | ${Z}_{7,1}=\overline{{Z}_{7,2}}$ |

Evaluation Indicator | Changning (E_{1}) | Weiyuan (E_{2}) | Fushun-Yongchuan (E_{3}) | Jiaoshiba (E_{4}) |
---|---|---|---|---|

Thickness of organic-rich shale (X_{17}) | 33–46 | 40–50 | 60–100 | 38–44 |

Total organic carbon (X_{18}) | 1.9–7.3/4.0 | 1.9–6.4/2.7 | 1.6–6.8/3.8 | 1.5–6.1/3.5 |

Organic maturity (X_{19}) | 2.6 | 2.7 | 2.5–3.0 | 2.6 |

Favorable area (X_{8}) | 2050 | 4216 | 3900 | 5450 |

Gas content (X_{9}) | 4.1 | 2.92 | 3.6 | 3.5 |

Tectonic setting (X_{10}) | Slope | Slope | Syncline negative structure | Anticline normal structure |

Burial depth (X_{11}) | 2.3–3.2 | 1.3–3.7 | 3.2–4.5 | 2.4–3.5 |

Pressure factor (X_{12}) | 1.35–2.03 | 0.92–1.77 | 2.0–2.25 | 1.35–1.55 |

Degree of natural cracks (X_{13}) | Interlayer seam development | Interlayer seam development | Full development | Full development |

Surface condition (X_{14}) | Mountains or dams | Mountains or dams | Plains or hills | Mountains or dams |

Market demand (X_{15}) | Medium | Larger | Medium | Larger |

Infrastructure (X_{16}) | Close to a pipe network | Close to a pipe network | Close to a pipe network | In a pipe network |

**Table 6.**Evaluation results of Table 5.

Area | Target Area (Level I) | Favorable Area (Level II) | Prospective Area (Level III) |
---|---|---|---|

Jiaoshiba | 100% | ||

Changning | 100% | ||

Fushun-Yongchuan | 51.51% | 48.49% | |

Weiyuan | 41.49% | 58.51% |

Area | Score | Rank | Level |
---|---|---|---|

Jiaoshiba | 72.7670 | 1 | I |

Changning | 72.2571 | 2 | I |

Fushun-Yongchuan | 71.3968 | 3 | II |

Weiyuan | 67.9926 | 4 | III |

Evaluation Indicator | Variable | Data |
---|---|---|

Thickness of organic-rich shale | X_{17} | 60–100 |

Total organic carbon | X_{18} | 1.6–6.8/3.8 |

Organic maturity | X_{19} | 2.5–3.0 |

Favorable area | X_{8} | 3900 |

Gas content | X_{9} | 3.6 |

Tectonic setting | X_{10} | Syncline negative structure |

Burial depth | X_{11} | 3.2–4.5 |

Pressure factor | X_{12} | — |

Degree of natural cracks | X_{13} | Fully developed |

Surface condition | X_{14} | Plains and hills |

Market demand | X_{15} | Balance between supply and demand |

Infrastructure | X_{16} | Close to a pipe network |

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## Share and Cite

**MDPI and ACS Style**

Yao, Q.; Yang, B.; Zhang, Q.
Dynamic Uncertain Causality Graph Applied to the Intelligent Evaluation of a Shale-Gas Sweet Spot. *Energies* **2021**, *14*, 5228.
https://doi.org/10.3390/en14175228

**AMA Style**

Yao Q, Yang B, Zhang Q.
Dynamic Uncertain Causality Graph Applied to the Intelligent Evaluation of a Shale-Gas Sweet Spot. *Energies*. 2021; 14(17):5228.
https://doi.org/10.3390/en14175228

**Chicago/Turabian Style**

Yao, Quanying, Bo Yang, and Qin Zhang.
2021. "Dynamic Uncertain Causality Graph Applied to the Intelligent Evaluation of a Shale-Gas Sweet Spot" *Energies* 14, no. 17: 5228.
https://doi.org/10.3390/en14175228