Krishna P.Paudel *Louisiana State University, Baton Rouge, USA *kpaudel@agcenter.lsu.edu

Biswo Poudel, Kathmandu University, Nepal

Fabien Prieur, INRA, France

Natural gas production through hydraulic fracturing (specifically horizontal slickwater fracking) since 1998 has brought or is likely to bring economic development in many parts of the U.S. Hydraulic fracturing has been the subject of much controversy and discussion because of its impact on groundwater quality, groundwater quantity, environmental quality, and health. Drilling of the vertical and horizontal components of a hydraulic fracturing well may require 400–4000 m^{3} (1 m^{3}=264.1 gallons or 35.3 ft^{3}) of water during the drilling phase. At the production phase, each hydraulic fracturing well requires about 7000–18,000 m^{3} of water. During the flowback period which generates wastewater known as produced water, which usually lasts up to two weeks, approximately 10 to 40% of the fracturing fluid (~99% water) returns to the surface (Arthur et al. 2008). Once active gas production has begun, water and other liquid continues to be produced at the surface in much lower volumes (2–8 m^{3}/day) over the lifetime of the well. There is a potential risk of groundwater over extraction from fracking operations in many aquifers (e.g. Louisiana’s Haynesville Shale and Wilcox aquifer). Similarly, the flow back water can get to waterbodies and render surface water toxic or non-useful for many aquatic species. To alleviate the groundwater depletion problem, some states (such as Louisiana) have asked fracking well operators to utilize surface water. This adds economic burden to fracking well operators as water needs to be transported from surface water source rather than drilling from the aquifer at the fracking station. We develop a theoretical model and then test our model using water use situation by fracking wells in Louisiana.

This paper is one of the first few papers to analyze the effects of hydraulic fracturing on water resources in a game theoretical framework. We developed a differential game to analyze the transboundary exploitation of groundwater for fracking in the presence of the risk of a catastrophic event that forces shift from groundwater to surface water and imposes a sunk cost to the users. Risk in our paper is broadly defined: it can be a catastrophic salt water intrusion due to over exploitation of groundwater aquifer or a legislative move to ban groundwater use in fracking. We also allow users to switch voluntarily to surface water at a lower cost, in order to protect themselves against the event occurrence. We analyze both the centralized solution (CS) as well as the symmetric Markov Perfect Nash equilibrium (MPE). We characterize a condition under which CS extraction rates are lower and stock and survival probability are higher than those in MPE. We also characterize the situation under which players switch to surface water sooner in MPE than in CS, contrary to what is generally expected in environmental economics literature. We conclude that strategic interaction may induce players to be more prudent than they would be in CS.

Let’s consider N ≥ 2 players, indexed by i = 1, …, N, who extract water in a common aquifer whose stock is denoted by Q. Let x_{i} be player i’s extraction. The amount of water extracted at each instant have different uses including municipal water for human consumption, irrigation water for agriculture use and industrial use including fracking. Benefits from extraction x_{i} are given by π_{i}(x_{i}) =a x_{i} (b− x_{i}). The state equation describing the evolution of the stock of groundwater is:

with h the exogenous constant recharge rate and Q(0) = Q_{0} given.

Extraction is subject to a risk of costly. Let’s for simplicity consider that once the event occurs, at some date τ , extraction from groundwater must stop and players have to switch to the alternative (more costly) source of water, namely surface water (lakes, river).

Besides the risk of salt intrusion that surrounds intensive use of groundwater, we want to account for all the potential problems created by the use of water for fracking (fracking may cause the contamination of soils and water by pollutants like oil, heavy metals etc.). We may also relate the event to the (unanticipated) implementation of a regulation that forces users to switch from groundwater to surface water in order to protect it from the occurrence of an irreversible and costly degradation.

In terms of modeling, it means that we can’t rely on the traditional approach to model water management under uncertainty. For instance, Tsur and Zemel (1995) offer the following description of the problem of groundwater management under a risk of catastrophic event (like salt intrusion). Basically during phases when total extraction exceeds the recharge rate, the stock of resource decreases. This decrease makes it likely the crossing of an unknown threshold that triggers the catastrophe. In addition, the only information available about the threshold is summarized by a probability distribution function with given density and support. We have to go beyond this description because the use of water for fracking exposes the system to a risk as long as total extraction is positive (no need to exceed the recharge rate).

Because of that specificity, we adapt the approach initially developed by Kamien and Schwartz (1971) to account for the risk of occurrence of costly events. It boils down as usual to define the date of the event τ as a random variable with probability distribution function F_{τ }(t) = prob(τ < t) deﬁned over the support [0, ∞), and density f_{τ} (t). The important assumption is that water withdrawals, x_{i}, by all players change the cumulative probability in the following way

So, the approach consists in introducing another state variable, hereafter we will work with the survival probability, 1 − F_{τ} (t). This variable is intimately linked with the stock of resource as it is apparent when combining the above two equations, we get:

Hereafter, we assume that all the players share the same information about the survival probability and its relation with the stock of resource. The information is provided to all users by some experts of the aquifer. Due to space limitation, we cannot provide the detail solutions. However, we present the two propositions which we have derived the proofs as well:

**Proposition 1**: *For CS: Under this condition, groundwater will be extracted till the horizon, i.e. T = Z, the extraction rate follows a monotone increasing trajectory and the stock of resource decreases monotonically if h < Nx. The survival probability is also monotone decreasing.*

**Proposition 2**: *At the MPE, players facing a risk of costly event choose to extract groundwater but they never switch to surface water and the equilibrium has the same feature as the centralized solution. In particular extraction rates are monotone increasing.*